In the English language, the word "magic" is and has always been a vague term. To refer to Superman as a "witch" would be a normal thing to do in many reallife contexts.
During Middle English, the term "magic" deriving from the biblical "magi" becoming a popular synonym for "fairie". The term "fairie" means magic, including the sense of a noun, "the activity of the fey", and adjective, "relating to the fey". Where fairie means a creature of "the realm of fey", it originally meant any magical creature, but came more and more to mean a specific species of magical creature, namely the fairy. Thus the use of the term "magic" helped disambiguate the intended meaning.
It helps to think of the D&D sources as different theories of magic: Arcane, Divine, Primal, and Psionic. Psionic is one of these sources.
Any source can:
• Produce effects that are spells
• Produce effects that are magic but not spells
• Magically produce effects that in themselves are nonmagical
Any source can use any kind of mechanic, whether spell slots, spell points, or other class feature mechanic.
When I think of the Psionic source, I view the "soul" (including aura, self, and conscious mind) to be natural and nonmagical, but able to magically produce effects, because the nature of magic itself is responsive to the intentions of the mind.